Flow Resistance over Mobile Bed in an Open-Channel Flow

This paper deals with the underlying mechanism of flow resistance in an alluvial channel: The effects of sidewall and bed form on flow resistance, Einstein’s divided hydraulic radius approach and Engelund’s energy slope division approach are reexamined. These two approaches assume that the shear stress on a mobile bed is the summation of shear stresses caused by skin friction and bed form. Using a different approach, this paper presents a theoretical relationship between the total bed shear stress with grain and bed-form shear stresses. The contribution of sidewall on the total bed shear stress is also discussed. The writers found that the size of bed form plays a significant role for the flow resistance, and developed relevant expressions for the length of the separation zone behind the bed forms. In addition, a systematical approach has been developed to compute the flow velocity in an alluvial channel. This approach is tested and verified against 5,989 flume and field measurements. The computed and measured discharge/velocity are in good agreement and 83.0% of all data sets fall within the ±20% error band.     

Bernoulli Theorem, Minimum Specific Energy, and Water Wave Celerity in Open-Channel Flow

One basic principle of fluid mechanics used to resolve practical problems in hydraulic engineering is the Bernoulli theorem along a streamline, deduced from the work-energy form of the Euler equation along a streamline. Some confusion exists about the applicability of the Bernoulli theorem and its generalization to open-channel hydraulics. In the present work, a detailed analysis of the Bernoulli theorem and its extension to flow in open channels are developed. The generalized depth-averaged Bernoulli theorem is proposed and it has been proved that the depth-averaged specific energy reaches a minimum in converging accelerating free surface flow over weirs and flumes. Further, in general, a channel control with minimum specific energy in curvilinear flow is not isolated from water waves, as customary state in open-channel hydraulics.     

Reynolds Stress and Bed Shear in Nonuniform Unsteady Open-Channel Flow

Expressions for the Reynolds stress and bed shear stress are developed for nonuniform unsteady flow in open channels with streamwise sloping beds, assuming universal (logarithmic) velocity distribution law and using the Reynolds and continuity equations of two-dimensional open-channel flow. The computed Reynolds stress distributions are in agreement with experimental data.     

Subcritical Contraction for Improved Open-Channel Flow Measurement Accuracy with an Upward-Looking ADVM

Acoustic Doppler velocity meters (ADVMs) provide an alternative to more traditional flow measurement devices and procedures such as flumes, weirs, and stage rating for irrigation and drainage canals. However, the requirements for correct calibration are extensive and complex. A three-dimensional computational fluid dynamics (CFD) model was used to design a subcritical rapidly varied flow contraction that provides a consistent linear relationship between the upward-looking ADVM sample velocity and the cross-sectional average velocity in order to improve ADVM accuracy without the need for in situ calibration. CFD simulations validated the subcritical contraction in a rectangular and trapezoidal cross section by showing errors within +1.8 and ?2.2%. Physical testing of the subcritical contraction coupled with an upward-looking ADVM in a large rectangular flume provided laboratory validation with measurement errors within ±4% without calibration.     

Flow in Open-Channel Embayments

This study investigates flows in a square and a rectangular embayment located on the side bank of an open channel. It is found that the main flow of the open channel induces a circulatory flow in the square and rectangular embayment and the center of the circulatory flow is shifted downstream in comparison with the center of the embayment. A solution of the shallow water equations solved using the method of variation yields results of the 95% confidence intervals within 10% of mean errors between the observed and computed nondimensional velocities.     

General Characteristics of Solutions to the Open-Channel Flow, Feedforward Control Problem

A dimensionless formulation of the open-channel flow equations was used to study the feedforward control problem for single-pool canals. Feedforward inflow schedules were computed for specified downstream demands using a gate-stroking model. The analysis was conducted for various design and operational conditions. Differences in the shape of the computed inflow hydrographs are largely related to the volume change resulting from the transient, the time needed to supply this volume, and the time needed by the inflow perturbation to travel down the canal. The gate-stroking method will fail to produce a solution or the solution will demand extreme and unrealistic inflow variations if the time needed to supply the canal volume change is much greater than the travel time of the upstream flow change. As an alternative, a simple feedforward-control flow schedule can be developed based on this volume change and a reasonable delay estimate. This volume compensating schedule can deliver the requested flow change and keep water levels reasonably close to the target under the range of conditions tested.     

Brief Analysis of Shallow Water Equations Suitability to Numerically Simulate Supercritical Flow in Sharp Bends

This work deals with the suitability of two-dimensional shallow water equations for the numerical simulation of supercritical free surface flows in bends, when the usual hypothesis of small width/curvature radius ratio does not hold. Here, a very reliable and accurate finite-volume, Godunov-type scheme is adopted for the numerical integration of the governing equations. Comparison with a selected set of experimental laboratory data and asymptotic analytical solutions shows that several aspects concerning the physics of the phenomenon are well reproduced, such as the blocking of the stream when the Froude number of the undisturbed flow is not large enough and the bend is sufficiently sharp, while maximum water depth in the bend is systematically underestimated.     

Reynolds Stress Anisotropy in Open-Channel Flow

This paper reports the results of an experimental study characterizing turbulence and turbulence anisotropy in smooth and rough shallow open-channel flows. The rough bed consists of a train of two-dimensional transverse square ribs with a ratio of the roughness height (k) to the total depth of flow (d) equal to 0.10. Three rib separations (p/k) of 4.5, 9, and 18 were examined. Here, p is the pitch between consecutive roughness elements and was varied to reproduce the classical condition of d- and k-type roughness. For each case, two-component velocity measurements were obtained using a laser Doppler velocimetry system at two locations for p/k = 4.5 and 9: on the top of the rib and above the cavity, and an additional location for p/k = 18. The measurements allow examination of the local variations of the higher-order turbulent moments, stress ratios as well as turbulence anisotropy. Large variations of the turbulence intensities, Reynolds shear stress, turbulent kinetic energy and turbulence production are found for y1<3k. In this region, the flow is more directly influenced by the shear layers from the preceding ribs. The higher-order moments appear to be similar for all rough surfaces beyond y1 ≈ 7k. In the outer layer (y1>3k), all higher-order turbulent moments for the k-type roughness show a substantial increase due to the complex interactions between the roughness and the remnants overlying shear layers shed from succeeding ribs. Analysis of the components of the Reynolds stress anisotropy tensor shows that at p/k = 18, the flow at y1<5k tends to be more isotropic which implies that for this particular case, the effect of the roughness density could also be important. On the smooth bed, at the shallower depths, the correlation coefficient near the free surface increases and turbulence tends to become less anisotropic.     

Hydraulic Radius for Evaluating Resistance Induced by Simulated Emergent Vegetation in Open-Channel Flows

The resistance induced by simulated emergent vegetation in open-channel flows has been interpreted differently in the literature, largely attributable to inconsistent uses of velocity and length scales in the definition of friction factor or drag coefficient and Reynolds number. By drawing analogies between pipe flows and vegetated channel flows, this study proposes a new friction function with the Reynolds number that is redefined by using a vegetation-related hydraulic radius. The new relationship is useful for consolidating various experimental data across a wide range of vegetation density. The results clearly show a monotonic decrease of the drag coefficient with the new Reynolds number, which is qualitatively comparable to other drag coefficient relationships for nonvegetated flows. This study also proposes a procedure for correcting sidewall and bed effects in the evaluation of vegetation drag.     

Maximum Velocity and Regularities in Open-Channel Flow

Maximum velocity in a channel section often occurs below the water surface. Its location is linked to the ratio of the mean and maximum velocities, velocity distribution parameter, location of mean velocity, energy and momentum coefficients, and probability density function underpinning a velocity distribution equation derived by applying the probability and entropy concepts. The mean value of the ratio of the mean and maximum velocities at a given channel section is stable and constant, and invariant with time and discharge. Its relationship with the others in turn leads to formation of a network of related constants that represent regularities in open-channel flows and can be used to ease discharge measurements and other tasks in hydraulic engineering. Under the probability concept, the ratio of mean and maximum velocities being constant means that the probability distribution underpinning the velocity distribution and other related variables is resilient, and that the same probability distribution is governing various phenomena observable at a channel section and explains the regularities in open-channel flows.     

Nonlinear Observer-Based Feedback for Open-Channel Level Control

This paper is devoted to nonlinear observer and controller design for water level control of open-channel flow in irrigation canals or dam-river systems. A finite-dimensional model, previously developed by orthogonal collocation methods, based on Saint Venant equations and used for control design, is now further used for online flow rate and water infiltration estimation. This is done by a so-called state observer. In particular, the estimates obtained in this way can successfully be used in a controller previously proposed, resulting in a water level control law using only two level measurements along the canal (instead of the four measurements previously needed). The study is restricted to the case of a rectangular wetted section and subcritical flow. The results have been validated by simulations, on an implicit finite difference simulator based on a Preissmann scheme for various scenarios.     

Critical Depth Relationships in Developing Open-Channel Flow

Doubts have been expressed about the validity of the critical depth defined in terms of the minimum specific energy head of the free-surface streamline when dealing with developing open-channel flows. This note examines the two approaches for defining critical flow, that based on the minimum specific energy of the free-surface streamline and that based on the mean energy head of the whole flow section. Large differences for the dimensionless critical depths are obtained with the two methods due to each critical depth proving to be a different control point on the free-surface profile. It is argued that both methods are different alternatives, although the critical depth concept was different in each case. Theoretical support to critical flow computations based on the free streamline is provided. An alternative approach for computing the discharge characteristics of broad-crested weirs based on the energy loss inside the boundary layer is also proposed.     

Influence of Coherent Flow Structures on the Dynamics of Suspended Sediment Transport in Open-Channel Flow

The influence of suspended sediments on coherent flow structures has been studied by simultaneously measuring the longitudinal and vertical components of the instantaneous velocity vector and the instantaneous suspended particle concentration with an acoustic particle flux profiler. The measurements were carried out in clear water and in particle-laden open-channel flows. In both cases, they clearly show the predominance of ejection and sweep phases that are part of a burst cycle. The analysis further demonstrates the importance of the ejection and sweep phases in sediment resuspension and transport. Ejections pick up the sediment at the bed and carry it up through the water column close to the surface. It is shown that ejections and sweeps are in near equality in the near-bottom layer, whereas ejections clearly dominate in the remaining water column. The implications of these results for sediment transport dynamics are discussed.     

Discontinuous Galerkin Finite-Element Method for Transcritical Two-Dimensional Shallow Water Flows

A total variation diminishing Runge Kutta discontinuous Galerkin finite-element method for two-dimensional depth-averaged shallow water equations has been developed. The scheme is well suited to handle complicated geometries and requires a simple treatment of boundary conditions and source terms to obtain high-order accuracy. The explicit time integration, together with the use of orthogonal shape functions, makes the method for the investigated flows computationally as efficient as comparable finite-volume schemes. For smooth parts of the solution, the scheme is second order for linear elements and third order for quadratic shape functions both in time and space. Shocks are usually captured within only two elements. Several steady transcritical and transient flows are investigated to confirm the accuracy and convergence of the scheme. The results show excellent agreement with analytical solutions. For investigating a flume experiment of supercritical open-channel flow, the method allows very good decoupling of the numerical and mathematical model, resulting in a nearly grid-independent solution. The simulation of an actual dam break shows the applicability of the scheme to nontrivial bathymetry and wave propagation on a dry bed.     

Analytical Solutions for Open-Channel Temperature Response to Unsteady Thermal Discharge and Boundary Heating

Analytical solutions are derived for a one-dimensional model of the bulk temperature response of open-channel flow with unsteady and nonuniform heating at the upstream end, the water surface, and the riverbed. The solutions are explicit formulas comprised of transient terms, which play dominant roles in the upstream region, and equilibrium terms, which determine the temperature far downstream. The applicability of the solutions to practical problems is illustrated for two cases: (1) a stream bounded at its upstream end by a dam and with a midreach inflow; and (2) Boulder Creek, Colo., which is impacted by effluent released from a wastewater treatment plant. The model prediction is in reasonable agreement with gauged data.     

New Approach for Predicting Flow Bifurcation at Right-Angled Open-Channel Junction

An unsteady mathematical model for predicting flow divisions at a right-angled open-channel junction is presented. Existing dividing models depend on a prior knowledge of a constant flow regime. In addition, their strong nonlinearity does not guarantee compatibility with the St. Venant solutions in the context of an internal boundary condition treatment. Assuming zero crest height at the junction region, a side weir model explicitly introduced within the one-dimensional St. Venant equations is used to cope with the two-dimensional pattern of the flow. An upwind implicit numerical solver is employed to compute the new governing equations. The performance of the proposed technique in predicting super-, trans-, and subcritical flow bifurcations is illustrated by comparing with experimental data and/or theoretical predictions. In all the tests, lateral-to-upstream discharge ratios (Rq) are successfully reproduced by the present technique with a maximum error magnitude of less than 9%.     

Parameter Estimation for Flow in Open-Channel Networks

Two mathematical models for parameter estimation in closed-loop, open-channel flow networks are presented. The parameter estimation models seek to determine the parameter values that would reproduce an observed flow profile in the channel network. The governing equations for gradually varied flow in channel networks are the optimization models’ constraints. The projected augmented Lagrangian method is used to solve the optimization models. Performance of these two optimization models is evaluated for a given closed-loop network configuration. Results establish the potential of the developed models for use in real-life flow scenarios.     

Classifying River Waves by the Saint Venant Equations Decoupled in the Laplacian Frequency Domain

The Saint Venant equations are often combined into a single equation for ease of solution. As a result however, this single equation gives rise to several redundant nonlinear terms that may impose significant limitations on model analyses. In order to avoid this, our paper employs a new procedure that separates, in the Laplace frequency domain, the governing equation of water depth from that of flow velocity and thus enables us to consider two independent equations rather than two coupled ones. The so-obtained analytical solutions are valid for prismatic channels of any shape. Solution validity is assured by repeated comparison with the corresponding numerical solutions based on Crump’s algorithm, which accelerates solution convergence. Utilizing this new procedure, this paper will construct a basic wave spectrum for classifying subcritical flow waves in a prismatic channel. The spectrum is basically a contour plot of the normalized specific energy loss for a small water wave moving in the channel for a finite distance of approximately 100?m. The distance is chosen so that four distinct regions with different contour patterns that represent kinematic, diffusion, gravity, and dynamic waves in a river are shown in the spectrum. By incorporating the spectrum with Ferrick’s criteria and Manning’s formula, a single contour line is also generated, which serves as the boundary of the four regions. Example computations show that the spectrum predicts a similar trend of wave attenuation for waves propagating in a trapezoidal channel. When the rising speed of a wave is of concern, the full Saint Venant equations are solved numerically to reconstruct a similar spectrum good for supercritical flow as well.     

Physical and Numerical Comparison of Flow over Ogee Spillway in the Presence of Tailwater

Data obtained from two physical models were compared to the results obtained from numerical model investigations of two ogee-crested spillways. In 2001, Savage and Johnson investigated ogee-crested spillways without the effect of tailwater; the present study includes the influence of tailwater on the spillway. The comparison showed that numerical modeling can accurately predict the rate of flow over the spillway and the pressure distribution on the spillway. The results of this study provide users of numerical models performance information that can be used to aid them in determining which tool to use to effectively analyze dams and their associated spillways.     

Use of Stereoscopy for Dam Break Flow Measurement

This investigation explored the applicability of video stereoscopy for the measurement of unsteady open channel flows. Specifically, the three-dimensional water surface profile and flow velocities associated with scale model dam break events were considered. Stereo images of the unsteady flow event were obtained using three, time-synchronized, video cameras situated above the tank such that, at all times, the area of interest was captured by at least two of the three cameras. To establish a point of reference from image to image, floating plastic tracking particles were placed on the water surface. The three-dimensional coordinates of the particles were then calculated using the camera positions and the locations of the individual plastic particles in the stereo images. Particle velocities were also deduced from the analysis of consecutive images. Based on this preliminary investigation we conclude that video stereoscopy is a promising method for measuring highly dynamic flows.